Resonance method for measuring water-oil ratio, conductivity, porosity, permeability and electrokinetic constant in porous formations

ABSTRACT

A shear wave is generated at a borehole wall. A static magnetic field is applied with a radial direction and an oscillating magnetic field is applied tangential to the borehole wall. The frequency of the oscillating field is varied until a resonance condition occurs. Motion of the formation under resonance provides an indication of a resistivity property of the earth formation.

BACKGROUND OF THE PRESENT DISCLOSURE

1. Field of the Disclosure

The present disclosure relates generally to geological exploration inwellbores. More particularly, the present disclosure describes anapparatus, a machine-readable medium, and a method useful for usingacoustic measurements made in magnetic fields for determining formationproperties.

2. Description of the Related Art

A variety of techniques are currently utilized in determining thepresence and estimating quantities of hydrocarbons (oil and gas) inearth formations. These methods are designed to determine formationparameters, including, among other things, the resistivity, porosity,and permeability of a rock formation surrounding a wellbore drilled forrecovering the hydrocarbons. Typically, the tools designed to providethe desired information are used to log the wellbore. Much of thelogging is done after the wellbores have been drilled.

Extensive work has been done in the determination of formationproperties using nuclear magnetic resonance (NMR) methods. In the NMRmethod, a magnetic field is applied to formation which aligns thenuclear spins in a direction parallel to the magnetic field. Theformation is then pulsed with a pulsed radio frequency magnetic fieldorthogonal to the static magnetic field which changes the direction ofthe nuclear spins. Signals resulting from precession of the nuclearspins are measured, and with proper selection of the pulsing parameters,various formation properties such as porosity and diffusivity can bemeasured. There has been little recognition of measurements other thanNMR measurements that can be made in magnetic fields. U.S. patentapplication Ser. No. 11/696,461 of Tabarovsky et al., having the sameassignee as the present disclosure and the contents of which areincorporated herein by reference, teaches and claims a method ofdetermining a resistivity parameter of an earth formation through casingusing acoustic measurements in crossed-magnetic fields. U.S. patentapplication Ser. No. 11/847,920 of Dorovsky et al., having the sameassignee as the present disclosure and the contents of which areincorporated herein by reference discloses the determination offormation properties in crossed-magnetic fields. The present disclosureis related to determination of other properties using measurements inmagnetic fields.

SUMMARY OF THE DISCLOSURE

One embodiment of the disclosure is a method of estimating a resistivityproperty of an earth formation. The method includes generating a shearwave in an earth formation at an interface between a fluid and theformation; generating a static magnetic field having a first directionat the interface; varying a frequency of a second magnetic field havinga direction different from the first direction until a resonancecondition is satisfied in the formation; and using at least onemeasurement of motion of the earth formation to provide the estimate ofthe resistivity property.

Another embodiment of the disclosure is an apparatus configured toestimate a resistivity property of an earth formation. The apparatusincludes: a transducer configured to generate a shear wave in an earthformation at an interface between a fluid and the formation; a magnetarrangement configured to generate a static magnetic field at theinterface having a first direction and generate a time varying magneticfield at the interface in a second direction different from the firstdirection; and a processor configured to vary a frequency of thetime-varying magnetic field until a resonance condition is satisfied inthe formation use at least one measurement of motion of the earthformation to provide the estimate of the resistivity property.

BRIEF DESCRIPTION OF THE DRAWINGS

The present claimed subject matter may be better understood by referenceto one or more of these drawings wherein:

FIG. 1 shows an exemplary apparatus suitable for performing the methodof the present disclosure conveyed within a wellbore;

FIG. 2 schematically illustrates a magnet configuration of a resistivityapparatus suitable for use with the present disclosure; and

FIG. 3 shows the dependence of |u_(z)|/|B_(oz)(ω)| on frequency fordifferent values of the friction coefficient.

DETAILED DESCRIPTION OF THE DISCLOSURE

The present invention is discussed with reference to specific logginginstruments that may form part of a string of several logginginstruments for conducting wireline logging operations. It is to beunderstood that the choice of the specific instruments discussed hereinis not to be construed as a limitation and that the method of thepresent invention may also be used with other logging instruments aswell.

A typical configuration of the logging system is shown in FIG. 1. Thisis a modification of an arrangement from U.S. Pat. No. 4,953,399 toFertl et al., having the same assignee as the present invention, thecontents of which are incorporated herein by reference. Shown in FIG. 1is a suite of logging instruments 10, disposed within a borehole 11penetrating an earth formation 13, illustrated in vertical section, andcoupled to equipment at the earth's surface, in accordance with variousillustrative embodiments of the method and apparatus of the presentinvention. Logging instrument suite 10 may include a resistivity device12, a natural gamma ray device 14, and/or two porosity-determiningdevices, such as a neutron device 16 and/or a density device 18.Collectively, these devices and others used in the borehole for loggingoperations are referred to as formation evaluation sensors. Theresistivity device 12 may be one of a number of different types ofinstruments known to the art for measuring the electrical resistivity offormations surrounding a borehole so long as such device has arelatively deep depth of investigation. For example, a HDIL (HighDefinition Induction Logging) device such as that described in U.S. Pat.No. 5,452,761 to Beard et al., having the same assignee as the presentinvention, the contents of which are fully incorporated herein byreference, may be used. The natural gamma ray device 14 may be of a typeincluding a scintillation detector including a scintillation crystalcooperatively coupled to a photomultiplier tube such that when thecrystal is impinged by gamma rays a succession of electrical pulses isgenerated, such pulses having a magnitude proportional to the energy ofthe impinging gamma rays. The neutron device 16 may be one of severaltypes known to the art for using the response characteristics of theformation to neutron radiation to determine formation porosity. Such adevice is essentially responsive to the neutron-moderating properties ofthe formation. The density device 18 may be a conventional gamma-gammadensity instrument such as that described in U.S. Pat. No. 3,321,625 toWahl, used to determine the bulk density of the formation. A downholeprocessor 29 may be provided at a suitable location as part of theinstrument suite.

The logging instrument suite 10 is conveyed within borehole 11 by acable 20 containing electrical conductors (not illustrated) forcommunicating electrical signals between the logging instrument suite 10and the surface electronics, indicated generally at 22, located at theearth's surface. The logging devices 12, 14, 16, and/or 18 within thelogging instrument suite 10 are cooperatively coupled such thatelectrical signals may be communicated between each of the loggingdevices 12, 14, 16, and/or 18 and the surface electronics 22. The cable20 is attached to a drum 24 at the earth's surface in a manner familiarto the art. The logging instrument suite 10 is caused to traverse theborehole 11 by spooling the cable 20 on to or off of the drum 24, alsoin a manner familiar to the art.

The surface electronics 22 may include such electronic circuitry as isnecessary to operate the logging devices 12, 14, 16, and/or 18 withinthe logging instrument suite 10 and to process the data therefrom. Someof the processing may be done downhole. In particular, the processingneeded for making decisions on speeding up (discussed below) or slowingdown the logging speed is preferably done downhole. If such processingis done downhole, then telemetry of instructions to speed up or slowdown the logging could be carried out substantially in real time. Thisavoids potential delays that could occur if large quantities of datawere to be telemetered uphole for the processing needed to make thedecisions to alter the logging speed. It should be noted that withsufficiently fast communication rates, it makes no difference where thedecision-making is carried out. However, with present data ratesavailable on wirelines, the decision-making is preferably done downhole.

Control circuitry 26 contains such power supplies as are required foroperation of the chosen embodiments of logging devices 12, 14, 16,and/or 18 within the logging instrument suite 10 and further containssuch electronic circuitry as is necessary to process and normalize thesignals from such logging devices 12, 14, 16, and/or 18 in aconventional manner to yield generally continuous records, or logs, ofdata pertaining to the formations surrounding the borehole 11. Theselogs may then be electronically stored in a data storage 32 prior tofurther processing. A surface processor 28 may process the measurementsmade by the formation evaluation sensor(s) 12, 14, 16, and/or 18. Thisprocessing could also be done by the downhole processor 29.

The surface electronics 22 may also include such equipment as willfacilitate machine implementation of various illustrative embodiments ofthe method of the present invention. The surface processor 28 may be ofvarious forms, but preferably is an appropriate digital computerprogrammed to process data from the logging devices 12, 14, 16, and/or18. A memory unit 30 and the data storage unit 32 are each of a type tointerface cooperatively with the surface processor 28 and/or the controlcircuitry 26. A depth controller 34 determines the longitudinal movementof the logging instrument suite 10 within the borehole 11 andcommunicates a signal representative of such movement to the surfaceprocessor 28. The logging speed is altered in accordance with speedup orslowdown signals that may be communicated from the downhole processor29, and/or provided by the surface processor 28, as discussed below.This is done by altering the rotation speed of the drum 24. Offsitecommunication may be provided, for example, by a satellite link, by atelemetry unit 36.

Turning now to FIG. 2, an acoustic transmitter 203 is positioned at theborehole wall 11 in contact with the formation. 211. The formation isporous and saturated with an electrolyte. At the borehole wall 11, amulticomponent motion sensor (geophone or accelerometer) 205 ispositioned. In one embodiment of the disclosure, the magnetic fieldsensor may include a loop antenna. The acoustic transmitter 203 isactivated to produce in the porous medium two acoustic waves. In oneembodiment of the disclosure, the generated acoustic waves propagateprimarily along the axial direction of the borehole. This direction maybe referred to as the x-direction. The apparatus also includes a firstmagnet arrangement such as denoted by 201 a, 201 b that produces aconstant magnetic field in the formation with a radial direction. Asecond magnet, denoted by 221 provides a time-varying magnetic fieldthat is approximately tangential to the borehole. The analysis that ismade below uses a planar approximation for the interface. The coordinatesystem used is that the x-axis is in the radial direction (perpendicularto the interface between the fluid and the formation), and the y- andz-axes are along the borehole wall. For the purposes of the disclosure,it is sufficient that the magnetic field have a component along thedirection of propagation of the acoustic wave in the formation. Themagnets 201 a, 201 b, 221 are electromagnets driven by a variablefrequency source (not shown).

The acoustic transmitter is positioned in the plane x=0. Motion of thetransducer in the yz plane will produce a shear wave in the formation.In the absence of the magnetic field, this can be decomposed into twoindependent shear waves propagating into the formation, one withpolarization along the y-axis and one with polarization along thez-axis. In the presence of the static magnetic field and electrolytes inthe formation fluid, magnetoacoustic vibrations result in the formationresulting in two different electroacoustic waves. In one embodiment ofthe disclosure, the acoustic transmitter includes a wide-bandtransducer. The frequency of excitation of the transducer is changed,and the strength of the magnetic field in the magnetic field sensor as afunction of frequency is recorded. As discussed below, there is aresonance frequency at which the magnetic field exhibits a maximumamplitude. This resonance frequency is related to the formationproperties.

The theory behind the electroacoustic resonance method for measuringwater-oil ratio in a porous medium in the presence of electrolyte is theelectrodynamic theory combined with the non-linear theory of elasticdeformations of fluid-saturated media. Such a theory considers thefollowing conservation laws as the initial set of differentialequations:

Conservation of Mass:

$\begin{matrix}{{{\frac{\partial\rho}{\partial t} + {\nabla{\cdot \left( {{\rho_{s}u} + {\rho_{l}v}} \right)}}} = 0},} & (1)\end{matrix}$conservation of entropy, energy and momentum:

$\begin{matrix}{{{\frac{\partial S}{\partial t} + {\nabla{\cdot \left\lbrack {\frac{S}{\rho}\left( {{\rho_{s}u} + {\rho_{l}v}} \right)} \right\rbrack}}} = \frac{R}{T}},} & (2) \\{{{\frac{\partial E}{\partial T} + {\nabla{\cdot Q}}} = 0},} & (3) \\{{{{\frac{\partial}{\partial t}\left\lbrack {\left( {{\rho_{s}u} + {\rho_{l}v}} \right) + \frac{E \times H}{4\pi\; c_{e}}} \right\rbrack}_{i} + {\partial_{k}\Pi_{ik}}} = 0},} & (4)\end{matrix}$and the first law of Thermodynamics:

$\begin{matrix}{{dE}_{0} = {{TdS} + {\mu\; d\;\rho} + \left\lbrack {{u - v},{dj}_{0}} \right\rbrack + {\frac{h_{ik}}{2}d\;{g_{ik}.}}}} & (5)\end{matrix}$The evolution of the tensor of the deformation of the matrix of theearth formation is given by:

$\begin{matrix}{{{\frac{\partial g_{ik}}{\partial t} + {g_{ik}{\partial_{i}u_{j}}} + {g_{ij}{\partial_{k}u_{j}}} + {u_{j}{\partial_{j}g_{ik}}}} = 0},} & (6)\end{matrix}$the motion of the electrolyte in the porous matrix is given by:

$\begin{matrix}{{\frac{\partial v}{\partial t} + {\left( {v,\nabla} \right)v}} = {{- {\nabla\mu}} - {\frac{S}{\rho}{\nabla T}} + f + {f^{\partial}.}}} & (7)\end{matrix}$The full energy is given by

$\begin{matrix}{E = {E_{0} + {vj}_{0} + \frac{\rho\; v^{2}}{2} + {\frac{E^{2} + H^{2}}{8\pi}.}}} & (8)\end{matrix}$

In eqns. (1)-(8), the following notation is used: ρ, ρ₁, ρ_(s) aredensity of the porous saturated medium, partial density of theelectrolyte in pores, partial density of the matrix; S, E, Eo areentropy, energy, and internal energy per unit of volume of the medium;u, v are velocities of the matrix and of the electrolyte contained init; E, H are the electric and magnetic fields; μ, T are the chemicalpotential and temperature from the first principle of thermodynamics;j₀=ρ_(s)(u−v) is the invariant component of the momentum.

Full energy expression is obtained from the energy equation used in thehydrodynamic theory, extended to a two-velocity continuous medium. Theelectromagnetic theory is included up to a square-law accuracy for v/c.The motion eqn (7) for the liquid component contains the motion force onthe right-hand side. This motion force is linear with respect togradients of thermodynamic variables whose equilibrium values areconstant within the scale of the system. The physical essence of thistheory is determined by dependence of flows from thermodynamic variablesand can be uniquely defined by the following physical principles ofgeneral nature: invariance of these equations in regard to Galileotransformation, Minkovsky transformation for electromagnetic fields,conservation laws, and the second principle of thermodynamics fornon-equilibrium systems. Based on these principles, dependences of flowsand forces upon thermodynamic degrees of freedom can be determined.

The stress tensor follows the relation:

$\begin{matrix}{h_{ik} = {{{- \rho^{2}}\frac{\partial\left( {E_{0}/\rho} \right)}{\partial\rho}\delta_{ik}} - {2\rho\; g_{im}{\frac{\partial\left( {E_{0}/\rho} \right)}{\partial g_{mk}}.}}}} & (9)\end{matrix}$The density tensor of the momentum flow satisfies

$\begin{matrix}{{\Pi_{ik} = {{\rho_{s}u_{i}u_{k}} + {\rho_{l}v_{i}v_{k}} + {h_{ij}g_{jk}} - \frac{B_{i}B_{k}}{4\pi} - \frac{E_{i}E_{k}}{4\pi} + {\frac{E^{2} + H^{2}}{8\pi}\delta_{ik}}}},} & (10)\end{matrix}$where c is the electrodynamic constant (speed of light). Pressure isdetermined by the thermodynamic equation:p=−E ₀ +TS+μρ+(u−v,j ₀)  (11),for energy flow:

$\begin{matrix}{{Q = {{\left( {\mu + \frac{v^{2}}{2}} \right)j} + {\frac{TS}{\rho}j} + {u\left( {u,j_{0}} \right)} + {\frac{c}{4\pi}E \times B} + {u_{i}h_{k\; m}g_{m\; i}}}},} & (12)\end{matrix}$dissipative function:

$\begin{matrix}{{R = {{f^{\partial}\left( {{\rho\; u} - j} \right)} + {i_{0}\left( {E + {\frac{j^{\mathbb{e}}}{c\;\chi} \times B}} \right)}}},} & (13)\end{matrix}$volume density of reversible forces:

$\begin{matrix}{{f = {\frac{\chi_{l}}{\rho_{l}}\left( {E + {\frac{j^{\mathbb{e}}}{c\;\chi} \times B}} \right)}},} & (14)\end{matrix}$includes volumetric densities of free charges χ_(s), χ_(l), χ=χ_(l)which correspond to subsystems in which the following equations work:χ_(s)=σ_(s)σ⁻¹χ, χ_(l)=σ_(l)σ⁻¹χ. Here, σ_(l), σ_(s), σ=σ_(s)+σ_(l) aredensities associated with the matching subsystems.

The dissipative function allows linking the dissipative force f^(∂) andthe invariant component of the full current density j^(e)i ₀ =j ^(e) −χ _(s) u−χ _(l) v  (15)to thermodynamic forces by means of introducing kinetic phenomenologicalcoefficients□, □, and □

$\begin{matrix}{{f^{\partial} = {{\chi\left( {{\rho\; u} - j} \right)} + {\alpha_{12}\left( {E + {\frac{j^{\mathbb{e}}}{c^{e}\chi} \times B}} \right)}}},{i_{0} = {{\alpha_{21}\left( {{\rho\; u} - j} \right)} + {\sigma\left( {E + {\frac{j^{\mathbb{e}}}{c^{e}\chi} \times B}} \right)}}},{{\alpha_{12}(B)} = {{\alpha_{21}\left( {- B} \right)} = {\alpha.}}}} & {(16).}\end{matrix}$These equations are closed by Maxwell's equations (with no polarizationeffect considered):

$\begin{matrix}{{{\nabla{\times E}} = {{- \frac{1}{c}}\frac{\partial B}{\partial t}}},{{\nabla{\times B}} = {{{+ \frac{1}{c}}\frac{\partial E}{\partial t}} + {\frac{4\pi}{c}j^{\mathbb{e}}}}},{{\nabla{\cdot E}} = {4{\pi\chi}}},{{\nabla{\cdot B}} = 0.}} & (17)\end{matrix}$

These equations do not consider effects related to transfers ofcontaminating compounds concentrations. Also, kinetic effects of scalarand tensor nature are not included. In irreversible vector flowsthermoconductivity effects are neglected. The state equation can beobtained both arbitrarily and in Hook approximation by means ofexpansion of internal energy according to thermodynamic degrees offreedom up to cubic terms of the series. These equations describeelectrolyte filtration in a porous matrix undergoing arbitrary elasticdeformations. These equations describe the entire range of acoustic,electroacoustic, and magnetoacoustic waves allowed in the system.

In linear approximation for velocities of deformation and fluid motionand in quasistationary approximation of the electromagnetic field, theset of equations can be reduced to a set of three equations, with linearaccuracy:

$\begin{matrix}{{{\overset{¨}{u} - {c_{l}^{2}\Delta\; u} - {a_{1}{{\nabla\nabla} \cdot u}} + {a_{2}{{\nabla\nabla} \cdot v}} - {\frac{\sigma_{I}}{4{\pi\sigma\rho}_{0,s}}{\nabla{\times \overset{.}{B} \times B^{(0)}}}} + {\frac{\rho_{0,l}^{2}}{\rho_{0,s}}{\overset{\_}{\chi}\left( {\overset{.}{u} - \overset{.}{v}} \right)}} + {\frac{\alpha\; c\;\rho_{0,l}}{4{\pi\sigma\rho}_{0,s}}{\nabla{\times \overset{.}{B}}}}} = 0},{{\overset{¨}{v} - {a_{4}{{\nabla\nabla} \cdot v}} + {a_{3}{{\nabla\nabla} \cdot u}} - {\frac{\sigma_{l}}{4{\pi\sigma\rho}_{0,l}}{\nabla{\times \overset{.}{B} \times B^{(0)}}}} - {\rho_{0,l}{\overset{\_}{\chi}\left( {\overset{.}{u} - \overset{.}{v}} \right)}} - {\frac{\alpha\; c}{4{\pi\sigma}}{\nabla{\times \overset{.}{B}}}}} = 0},{\frac{\partial B}{\partial t} = {\nabla{\times {\left\lbrack {{{- \frac{c^{2}}{4{\pi\sigma}}}{\nabla{\times B}}} + \frac{\alpha\; c\;{\rho_{l}\left( {u - v} \right)}}{\sigma} + {\frac{\sigma_{s}}{\sigma}u \times B^{(0)}} + {\frac{\sigma_{l}}{\sigma}v \times B^{(0)}}} \right\rbrack.}}}}} & (18)\end{matrix}$Here, B⁽⁰⁾ is the external magnetic field, α_(i) are coefficientsrelated to elastic modules, □_(0,l) □_(0,s) are partial densities of thenon-excited medium, χ=χ−α²/σ, χ=η/(ρρ_(0,l)·k), k—permeability.

1D analysis in the absence of the external magnetic field of theelectroacoustic theory is related to propagation of electroacousticwaves along the x-axis

$\begin{matrix}{{{{\overset{¨}{u}}_{z} - {c_{l}^{2}\frac{\partial^{2}u_{z}}{\partial x^{2}}} + {\frac{\rho_{0,l}^{2}}{\rho_{0,s}}{\overset{\sim}{\chi}\left( {{\overset{.}{u}}_{z} - {\overset{.}{v}}_{z}} \right)}} + {\frac{a\; c\;\rho_{0,l}}{4{\pi\sigma\rho}_{0,s}}\frac{\partial{\overset{.}{B}}_{y}}{\partial x}}} = 0},{{{\overset{¨}{u}}_{y} - {c_{l}^{2}\frac{\partial^{2}u_{y}}{\partial x^{2}}} + {\frac{\rho_{0,l}^{2}}{\rho_{0,s}}{\overset{\sim}{\chi}\left( {{\overset{.}{u}}_{y} - {\overset{.}{v}}_{y}} \right)}} + {\frac{a\; c\;\rho_{0,l}}{4{\pi\sigma\rho}_{0,s}}\frac{\partial{\overset{.}{B}}_{z}}{\partial x}}} = 0},{{{\overset{¨}{v}}_{z} - {\rho_{0,l}{\overset{\sim}{\chi}\left( {{\overset{.}{u}}_{z} - {\overset{.}{v}}_{z}} \right)}} + {\frac{a\; c}{4{\pi\sigma}}\frac{\partial{\overset{.}{B}}_{y}}{\partial x}}} = 0},{{{\overset{¨}{v}}_{y} - {\rho_{0,l}{\overset{\sim}{\chi}\left( {{\overset{.}{u}}_{y} - {\overset{.}{v}}_{y}} \right)}} + {\frac{a\; c}{4{\pi\sigma}}\frac{\partial{\overset{.}{B}}_{z}}{\partial x}}} = 0},{\frac{\partial B_{y}}{\partial y} = {\frac{\partial}{\partial x}\left\lbrack {{\frac{c^{2}}{4{\pi\sigma}}\frac{\partial B_{y}}{\partial x}} - {\frac{a\; c\;\rho_{l}}{\sigma}\left( {u_{z} - v_{z}} \right)}} \right\rbrack}},{\frac{\partial B_{z}}{\partial y} = {{\frac{\partial}{\partial x}\left\lbrack {{\frac{c^{2}}{4{\pi\sigma}}\frac{\partial B_{z}}{\partial x}} - {\frac{a\; c\;\rho_{l}}{\sigma}\left( {u_{y} - v_{y}} \right)}} \right\rbrack}.}}} & (19)\end{matrix}$

It appears clear that, in the absence of the external magnetic field,two groups of independently propagating waves are observed. In thepresence of the magnetic field, interaction between the two independentmodes creates a polarized wave described by this set of equations:

$\begin{matrix}{{{{\overset{¨}{u}}_{z} - {c_{l}^{2}\frac{\partial^{2}u_{z}}{\partial x^{2}}} - {\frac{\sigma_{s}B_{x}^{(0)}}{4{\pi\sigma\rho}_{0,s}}\frac{\partial{\overset{.}{B}}_{z}}{\partial x}} + {{\frac{\rho_{0,l}^{2}}{\rho_{0,s}}\left\lbrack {\chi - \frac{\alpha^{2}}{\sigma}} \right\rbrack}\left( {{\overset{.}{u}}_{z} - {\overset{.}{v}}_{z}} \right)} + {\frac{\alpha\; c\;\rho_{0,l}}{4{\pi\sigma\rho}_{0,s}}\frac{\partial{\overset{.}{B}}_{y}}{\partial x}}} = 0}{{{\overset{¨}{v}}_{z} - {\frac{\sigma_{l}B^{(0)}}{4{\pi\sigma\rho}_{0,l}}\frac{\partial{\overset{.}{B}}_{z}}{\partial x}} - {\rho_{0,l}{\overset{\sim}{\chi}\left( {{\overset{.}{u}}_{z} - {\overset{.}{v}}_{z}} \right)}} - {\frac{a\; c}{4{\pi\sigma}}\frac{\partial{\overset{.}{B}}_{y}}{\partial x}}} = 0}{{{\overset{¨}{u}}_{y} - {c_{l}^{2}\frac{\partial^{2}u_{y}}{\partial x^{2}}} - {\frac{\sigma_{s}B_{x}^{(0)}}{4{\pi\sigma\rho}_{0,s}}\frac{\partial{\overset{.}{B}}_{y}}{\partial x}} + {\frac{\rho_{0,l}^{2}}{\rho_{0,s}}{\overset{\sim}{\chi}\left( {{\overset{.}{u}}_{y} - {\overset{.}{v}}_{y}} \right)}} - {\frac{\alpha\; c\;\rho_{0,l}}{4{\pi\sigma\rho}_{0,s}}\frac{\partial{\overset{.}{B}}_{z}}{\partial x}}} = 0}{{{\overset{¨}{v}}_{y} - {\frac{\sigma_{l}B^{(0)}}{4{\pi\sigma\rho}_{0,l}}\frac{\partial{\overset{.}{B}}_{y}}{\partial x}} - {\rho_{0,l}{\overset{\sim}{\chi}\left( {{\overset{.}{u}}_{y} - {\overset{.}{v}}_{y}} \right)}} + {\frac{a\; c}{4{\pi\sigma}}\frac{\partial{\overset{.}{B}}_{z}}{\partial x}}} = 0}{\frac{\partial B_{y}}{\partial t} = {\frac{\partial}{\partial x}\left\lbrack {{\frac{c^{2}}{4{\pi\sigma}}\frac{\partial B_{y}}{\partial x}} - {\frac{a\; c\;\rho_{l}}{\sigma}\left( {u_{z} - v_{z}} \right)} + {\frac{\sigma_{s}}{\sigma}u_{y}B_{x}^{(0)}} + {\frac{\sigma_{l}}{\sigma}v_{y}B_{x}^{(0)}}} \right\rbrack}}{\frac{\partial B_{z}}{\partial t} = {{\frac{\partial}{\partial x}\left\lbrack {{\frac{c^{2}}{4{\pi\sigma}}\frac{\partial B_{z}}{\partial x}} - {\frac{a\; c\;\rho_{l}}{\sigma}\left( {u_{y} - v_{y}} \right)} + {\frac{\sigma_{s}}{\sigma}u_{z}B_{x}^{(0)}} + {\frac{\sigma_{l}}{\sigma}v_{z}B_{x}^{(0)}}} \right\rbrack}.}}} & (20)\end{matrix}$

For periodic excitation,

$\left( {u_{z},v_{z},{B_{y};u_{y}},v_{y},B_{y}} \right) = {\left( {u_{z},v_{z},{B_{y};u_{y}},v_{y},B_{y}} \right){\mathbb{e}}^{\frac{{\mathbb{i}\omega}\; x}{\xi}}{{\mathbb{e}}^{{- {\mathbb{i}\omega}}\; t}.}}$This then gives

$\begin{matrix}{{B_{y} = {4\pi\frac{\rho_{0,l}}{\rho_{0,s}}\xi\frac{\Delta}{{a^{2}{c^{2}/\sigma^{2}}} - {\overset{\sim}{\chi\;}{B_{0}^{2}/\omega^{2}}}}\left( {{\frac{a\; c}{\sigma}u_{z}} - {\frac{{\mathbb{i}}\overset{\sim}{\chi}\; B_{0}}{\omega}u_{y}}} \right)}},} & (21) \\{{B_{z} = {{- 4}\pi\frac{\rho_{0,l}}{\rho_{0,s}}\xi\frac{\Delta}{{a^{2}{c^{2}/\sigma^{2}}} - {\overset{\sim}{\chi\;}{B_{0}^{2}/\omega^{2}}}}\left( {{\frac{a\; c}{\sigma}u_{y}} + {\frac{{\mathbb{i}}\overset{\sim}{\chi}\; B_{0}}{\omega}u_{z}}} \right)}},{where}} & (22) \\{{\Delta = {{\left( {1 - \frac{c_{l}^{2}}{\xi^{2}}} \right)\left( {1 + \frac{{\mathbb{i}\rho}_{0,l}\overset{\sim}{\chi}}{\omega}} \right)} + \frac{{\mathbb{i}\rho}_{0,l}^{2}\overset{\sim}{\chi}}{\rho_{0,s}\omega}}},{\overset{\sim}{\chi} = {\chi - {\frac{\alpha^{2}}{\sigma}.}}}} & (23)\end{matrix}$The velocity ξ of these dispersive waves satisfies the relation:

$\begin{matrix}{{\frac{\alpha^{2}c^{2}\rho_{0,l}^{2}}{\rho_{0,s}\sigma^{2}} - {\left( {\frac{\alpha^{2}c^{2}\rho_{0,l}}{\sigma^{2}} + \frac{B_{0}^{2}}{\rho_{0,l}}} \right)\left( {\frac{c_{l}^{2}}{\xi_{2}} - 1} \right)} + \frac{{\mathbb{i}\rho}_{0,l}\overset{\sim}{\chi}B_{0}^{2}}{\rho_{0,s}\omega}} = {4{\pi\xi}^{2}{{\Delta\left( {1 + \frac{{\mathbb{i}}\; c^{2}\omega}{4{\pi\sigma\xi}^{2}}} \right)}.}}} & (24)\end{matrix}$In these relations, c_(t) is the velocity of a shear wave in the mediumin the absence of excitation. The matrix conductivity is assumed to bezero.

Eqns. (21) and (22) lead to the conclusion that resonance occurs whenthe acoustic excitation is at a frequency

$\begin{matrix}{\omega_{0} = {\frac{B_{0}}{c}{\frac{\overset{\sim}{\chi}\sigma}{\alpha}.}}} & (25)\end{matrix}$Thus, the resonance frequency depends not only on the external magneticfield, but also on the three kinetic phenomena: conductivity σ,effective magnetic permeability and the electrokinetics constant α. Foran exemplary field B₀ of 10³ Gauss, σ=10⁹ 1/S, α=10⁶ cm^(3/2/()/(g.s)the resonance frequency is approximately 60 Hz. This is well within thecapability of an acoustic transducer in a borehole.

The source of the electroacoustic signal, which is the time-dependentmagnetic field orthogonal to wave propagation direction, is placed onthe borehole wall. On the same wall we also have a receiver measuringthe velocities of surface deformation in two mutually orthogonaldirections. In this case the receiver will register (we consider planegeometry) these amplitudes:

$\begin{matrix}{{u_{y} = {{\mathbb{i}}\frac{{ɛ\alpha}\; c_{e}}{4{\pi\sigma\omega}}\frac{\beta_{1}\beta_{2}}{1 + {{{\mathbb{i}}\left( {1 + ɛ} \right)}{\overset{\_}{\omega}/\omega}}}\frac{{\beta_{2}{\mathbb{e}}^{{- \beta_{1}}x}} - {\beta_{1}{\mathbb{e}}^{{- \beta_{2}}x}}}{\beta_{1}^{2} - \beta_{2}^{2}}{B_{0z}(\omega)}}},{u_{z} = {\frac{\beta_{1}\beta_{2}}{\beta_{2}^{2} - \beta_{1}^{2}}\frac{ɛ\; B_{0}\overset{\sim}{\chi}}{4{\pi\omega}^{2}}\frac{\left( {{\beta_{2}{\mathbb{e}}^{{- \beta_{1}}x}} - {\beta_{1}{\mathbb{e}}^{{- \beta_{2}}x}}} \right)}{1 + {{{\mathbb{i}}\left( {1 + ɛ} \right)}{\overset{\_}{\omega}/\omega}}}{{B_{0z}(\omega)}.}}}} & (26)\end{matrix}$At resonance frequency, on the surface (x=0) these amplitudes are thusrelated:

$u_{z} = {{\mathbb{i}}\frac{\omega_{0}}{\omega}u_{y}}$

The sensor of the velocities of matrix deformation at the boundarybetween the two media measures two components of the vector u=(u_(y),u_(z)) simultaneously. The frequency of the magnetic field to achieveequality of |u_(x)| and |u_(y)|. This is defined as a resonancecondition.

Under resonance condition,

$\begin{matrix}{\omega_{0} = {\frac{B_{0}}{c_{e}}{\frac{\overset{\_}{\chi}\sigma}{\alpha}.{Hence}}}} & (27) \\{\frac{\alpha}{\overset{\_}{\chi}\sigma} = {\frac{B_{0}}{c_{e}\omega_{0}}.}} & (28)\end{matrix}$The quantity on the right hand side of eqn. (28) is a known quantitysince B₀ and □₀ are measured quantities.

Along with a measurement of the resonance frequency and the magneticfield, the deformation velocities are also measured at the resonancefrequency.

$\begin{matrix}{u_{0} = {\frac{ɛ\overset{\_}{\chi}\; B_{0}}{4{{\pi\omega}_{0}^{2}\left\lbrack {1 + {{{\mathbb{i}}\left( {1 + ɛ} \right)}{\overset{\_}{\omega}/\omega_{0}}}} \right\rbrack}}\frac{\beta_{1}\beta_{2}}{\beta_{1} + \beta_{2}}{{B_{0z}\left( \omega_{0} \right)}.}}} & (29)\end{matrix}$The dispersion relation is

$\begin{matrix}{\beta^{4} + {\left( {\frac{\varphi_{1}}{c_{l}^{2}} + \frac{\mathbb{i}\omega}{\alpha_{1}} - \frac{\theta}{c_{t}^{2}\alpha_{1}}} \right)\beta^{2}} + {\frac{{\mathbb{i}\omega\varphi}_{1}}{{c_{t}^{2}\alpha_{1}} = 0}.}} & (30)\end{matrix}$At the resonance frequency, □=0, which gives the two roots of eqn (30)as:

$\begin{matrix}{{\beta_{2}^{2} = {{- \frac{\varphi_{1}}{c_{t}^{2}}} = {{- \frac{\omega_{0}^{2}}{c_{t}^{2}}}\left( {1 + {ɛ\frac{{\mathbb{i}}{\overset{\_}{\omega}/\omega_{0}}}{1 + {{\mathbb{i}}{\overset{\_}{\omega}/\omega_{0}}}}}} \right)}}}{\beta_{1}^{2} = {{- \frac{{\mathbb{i}\omega}_{0}}{\alpha_{1}}} = {\frac{4{{\pi\sigma\omega}_{o}^{2}/c_{e}^{2}}}{{\mathbb{i}} - {\frac{1}{1 + {{\mathbb{i}}{\overset{\_}{\omega}/\omega_{0}}}}\left( {\frac{\hat{\omega}}{\omega_{0}} + {\frac{B_{0}^{2}}{\rho_{0,l}c_{e}^{2}}\frac{\sigma}{\omega_{0}}}} \right)}}.}}}} & (31)\end{matrix}$

When the following conditions are satisfied:

$\begin{matrix}{{\frac{\overset{\_}{\omega}}{\omega_{0}} > 1},{\frac{\hat{\omega}}{\omega_{0}} < 1},{\frac{B_{0}^{2}}{\rho_{0,l}c_{e}^{2}} < \frac{\omega_{0}}{\sigma}},} & (32)\end{matrix}$we get the relations:

$\begin{matrix}{{{\frac{\alpha}{\overset{\_}{\chi}} = {{\sigma\frac{B_{0}}{c_{e}\omega_{0}}} = {const}}};}{{{\sqrt{\sigma} \approx \frac{\sqrt{4{\pi\omega}_{0}}\rho_{0}c_{e}{u_{0}}}{B_{0}{B_{0z}\left( \omega_{0} \right)}}} = {const}},{{\alpha/\chi} \approx \frac{{ɛ\zeta\rho}_{0}}{4{\pi \cdot \eta}}},}} & (33)\end{matrix}$where ε is the permittivity, η is the dynamic viscosity, ζ- isζ-potential. The first of the two equations is the same as eqn. (25).The second equation gives the conductivity □under the resonancecondition since all the quantities are known. The third equation givesthe relationship α/χ with ζ-potential. It is also possible to measureχ=χ−α²/σ (therefore permeability—k) at any frequency using the formulaabove for α=0.

In the case (α=0) equations (20) have only one the group of equations:

$\begin{matrix}{{{{\overset{¨}{u}}_{z} - {c_{t}^{2}\frac{\partial^{2}u_{z}}{\partial x^{2}}} + {ɛ{\overset{\_}{\omega}\left( {{\overset{.}{u}}_{z} - {\overset{.}{v}}_{z}} \right)}}} = 0},{{{\overset{¨}{v}}_{z} - {\frac{B_{0}}{4{\pi\rho}_{0,l}}\frac{\partial{\overset{.}{B}}_{z}}{\partial x}} - {\overset{\_}{\omega}\left( {{\overset{.}{u}}_{z} - {\overset{.}{v}}_{z}} \right)}} = 0},{\frac{\partial B_{z}}{\partial t} = {{\frac{c_{e}^{2}}{4{\pi\sigma}}\frac{\partial^{2}B_{z}}{\partial x^{2}}} + {B_{0}\frac{\partial v_{z}}{\partial x}}}},} & (34)\end{matrix}$which describe propagation of independent elastic S-waves excited by thequasistationary magnetic field. In these formulae the characteristicfrequency ω=ρ_(0,l)χ is introduced, as well as the ratioε=ρ_(0,l)/ρ_(0,s). Let us discuss the group of waves and basic featuresof plane harmonic waves:(u_(z),v_(z),B_(z))

(u_(z),v_(z),B_(z))·exp(−iωt),whose dependence on coordinates is described by this set of differentialequations:

$\begin{matrix}{{{{\omega^{2}u_{z}} + {c_{t}^{2}\frac{\partial^{2}u_{z}}{\partial x^{2}}} + {{\mathbb{i}ɛ}\overset{\_}{\omega}{\omega\left( {u_{z} - v_{z}} \right)}}} = 0},{{{\omega\; v_{z}} - {{\mathbb{i}}\frac{B_{0}}{4{\pi\rho}_{0,l}}\frac{\partial B_{z}}{\partial x}} - {{\mathbb{i}}{\overset{\_}{\omega}\left( {u_{z} - v_{z}} \right)}}} = 0},{{{\frac{c_{e}^{2}}{4{\pi\sigma}}\frac{\partial^{2}B_{z}}{\partial x^{2}}} + {B_{0}\frac{\partial v_{z}}{\partial x}} + {{\mathbb{i}\omega}\; B_{z}}} = 0.}} & (35)\end{matrix}$From the second equation of the set we can find the fluid velocity:

$\begin{matrix}{{v_{z} = {{\frac{\mathbb{i}}{\omega + {{\mathbb{i}}\overset{\_}{\omega}}}\frac{B_{0}}{4{\pi\rho}_{0,l}}\frac{\partial B_{z}}{\partial x}} + {\frac{{\mathbb{i}}\overset{\_}{\omega}}{\omega + {{\mathbb{i}}\overset{\_}{\omega}}}u_{z}}}},} & (36)\end{matrix}$and then write out set (35) as a set of these two equations:

$\begin{matrix}{{{\frac{\partial^{2}u_{z}}{\partial x^{2}} + {\frac{\omega^{2}}{c_{t}^{2}}\left( {1 + \frac{{\mathbb{i}ɛ}\overset{\_}{\omega}}{\omega + {{\mathbb{i}}\overset{\_}{\omega}}}} \right)u_{z}} + {\frac{{ɛ\omega}\overset{\_}{\omega}}{\left( {\omega + {{\mathbb{i}}\overset{\_}{\omega}}} \right)}\frac{B_{0}^{2}}{4{\pi\rho}_{0,l}c_{t}^{2}}\frac{\partial B}{\partial x}}} = 0},{{{\frac{{\mathbb{i}}\overset{\_}{\omega}}{\left( {\omega + {{\mathbb{i}}\overset{\_}{\omega}}} \right)}\frac{\partial u_{z}}{\partial x}} + {D\frac{\partial^{2}B}{\partial x^{2}}} + {{\mathbb{i}\omega}\; B}} = 0.}} & (37)\end{matrix}$In formula (37) a measureless magnetic field is defined:B_(z)=B_(o)B  (38)as well as this parameter:

$\begin{matrix}{D = {\frac{c_{e}^{2}}{4{\pi\sigma}} + {\frac{{\mathbb{i}}\; B_{0}^{2}}{4{{\pi\rho}_{0,l}\left( {\omega + {{\mathbb{i}}\overset{\_}{\omega}}} \right)}}.}}} & (39)\end{matrix}$

The latter equation serves as an equation which determines the velocityof the matrix deformation through the measureless magnetic field:

$\begin{matrix}{{\frac{\partial u_{z}}{\partial x} = {{- \left( {1 - {{\mathbb{i}}\frac{\omega}{\overset{\_}{\omega}}}} \right)}\mspace{11mu} D\hat{L}B}},} & (40)\end{matrix}$which, in its turn, is found from a linear differential equation of thefourth order:

$\begin{matrix}{{\hat{L}\left\lbrack {\frac{\partial^{2}B}{\partial x^{2}} + {\frac{\omega^{2}}{c_{t}^{2}}\left( {1 + \frac{ɛ\;{\mathbb{i}}\overset{\_}{\omega}}{\omega + {{\mathbb{i}}\overset{\_}{\omega}}}} \right)B}} \right\rbrack} = {\frac{B_{0}^{2}}{4{\pi\rho}_{0,1}c_{t}^{2}}\frac{{\mathbb{i}ɛ}{\overset{\_}{\omega}}^{2}\omega}{{D\left( {\omega + {{\mathbb{i}}\overset{\_}{\omega}}} \right)}^{2}}{\frac{\partial^{2}B}{\partial x^{2}}.}}} & (41)\end{matrix}$A linear differential operator is introduced in these equations:

$\begin{matrix}{\hat{L} = {\frac{\partial^{2}}{\partial x^{2}} + {\frac{\mathbb{i}\omega}{D}.}}} & (42)\end{matrix}$As we already have a solution to equation (41) for the magnetic field,deformation velocity of the porous matrix is found by integrating (40).For a boundless half-space x>0 exponentially attenuating solutions likethe one below may be of interest:B˜e^(−βx).  (43)Substituting (43) in equation (41), we arrive at a bi-quadraticalgebraic equation which determines possible values of the exponentialfactor.We would like to find the roots of this equation:

$\begin{matrix}{{{\left\lbrack {\beta^{2} + {\frac{\omega^{2}}{c_{t}^{2}}\left( {1 + \frac{{ɛ\mathbb{i}}\overset{\_}{\omega}}{\omega + {{\mathbb{i}}\overset{\_}{\omega}}}} \right)}} \right\rbrack\left\lbrack {\beta^{2} + {{\mathbb{i}}\frac{\omega}{D}}} \right\rbrack} = {\frac{B_{0}^{2}}{4{\pi\rho}_{0,1}c_{t}^{2}}\frac{{\mathbb{i}ɛ}{\overset{\_}{\omega}}^{2}\omega}{{D\left( {\omega + {{\mathbb{i}}\overset{\_}{\omega}}} \right)}^{2}}\beta^{2}}},} & (44)\end{matrix}$which contain positive real parts:β={β₁,β₂}, Reβ₁>0, Reβ₂>0.  (45)

The latter form of the roots is convenient for calculating them at smallvalues of the external longitudinal magnetic field B₀. Bi-quadraticequation (44) has two roots with positive real parts:B=N _(1z) e ^(−β) ¹ ^(x) +N _(2z) e ^(−β) ² ^(x).  (46)Equation (40) enables us to write

$\begin{matrix}{{u_{z} = {{\left( {1 - {{\mathbb{i}}\frac{\omega}{\overset{\_}{\omega}}}} \right)\frac{M_{1}}{\beta_{1}}N_{1z}{\mathbb{e}}^{{- \beta_{1}}x}} + {\left( {1 - {{\mathbb{i}}\frac{\omega}{\overset{\_}{\omega}}}} \right)\frac{M_{2}}{\beta_{2}}N_{2z}{\mathbb{e}}^{{- \beta_{2}}x}}}},{M_{1} = {{D\;\beta_{1}^{2}} + {\mathbb{i}\omega}}},{M_{2} = {{D\;\beta_{2}^{2}} + {{\mathbb{i}\omega}.}}}} & (47)\end{matrix}$Constants N_(1z), N_(2z) are found from boundary conditions:

$\begin{matrix}{{{N_{1z} + N_{2z}} = {{{B_{z}(\omega)}\left( {x = 0} \right)} = \frac{B_{z\; 0}(\omega)}{B_{0}}}},{{{{- \left( {1 - {{\mathbb{i}}\frac{\omega}{\overset{\_}{\omega}}}} \right)}M_{1}N_{1z}} - {\left( {1 - {{\mathbb{i}}\frac{\omega}{\overset{\_}{\omega}}}} \right)M_{2}N_{2z}}} = 0.}} & (48)\end{matrix}$

The first condition reflects the presence of the external time-dependentmagnetic field on the surface of the boundary x=0, the second conditionreflects the absence of tangential forces applied to the surface. Simplecalculations lead to:

$\begin{matrix}{{B_{z} = {\frac{B_{z\; 0}(\omega)}{D\left( {\beta_{1}^{2} - \beta_{2}^{2}} \right)}\left( {{M_{1}{\mathbb{e}}^{{- \beta_{2}}x}} - {M_{2}{\mathbb{e}}^{{- \beta_{1}}x}}} \right)}},{u_{z} = {\frac{B_{z\; 0}(\omega)}{{DB}_{0}}\left( {1 - {{\mathbb{i}}\frac{\omega}{\overset{\_}{\omega}}}} \right)\frac{M_{1}M_{2}}{\beta_{2}^{2} - \beta_{1}^{2}}{\left( {\frac{{\mathbb{e}}^{{- \beta_{1}}x}}{\beta_{1}} - \frac{{\mathbb{e}}^{{- \beta_{2}}x}}{\beta_{2}}} \right).}}}} & (49)\end{matrix}$The first solution determines how the external alternating magneticfield permeates through the porous medium, the second solutiondetermines the character of acoustic waves generated by it in thepresence of the longitudinal stationary magnetic field (longitudinalmeaning aligned with the direction of wave propagation). First of all,let us note that according to formula (11), two S-waves are generated insuch a situation. The superposition of these waves at the boundary x=0gives the deformation velocity of the matrix:

$\begin{matrix}{{u_{z}\left( {x = 0} \right)} = {\frac{B_{z\; 0}(\omega)}{{D(\chi)}B_{0}}\left( {1 - {{\mathbb{i}}\frac{\omega}{\overset{\_}{\omega}}}} \right)\frac{{M_{1}(\chi)}{M_{2}(\chi)}}{{\beta_{2}^{2}(\chi)} - {\beta_{1}^{2}(\chi)}}{\left( {\frac{1}{\beta_{1}(\chi)} - \frac{1}{\beta_{2}(\chi)}} \right).}}} & (50)\end{matrix}$Italics here show the exact solution and the equation which determinesroots β₁, β₂.

Let us consider this frequency dependence of |u_(z)|/|B_(oz)(ω)| forvarious values of the friction coefficient χ shown in FIG. 3. The curves301, 303 and 305 correspond to values of χ of 10⁴ cm³/(g.s), 10⁵cm³/(g.s) and 10⁶ cm³/(g.s) respectively. From this, it is clear that ifwe have adequate frequency dependence, we can estimate permeability ofthe system, regardless of what we know about conductivity. In this case,as it is seen from the graph, the resolution is higher in thehigher-frequency domain. Note that we need to know conductivity here ifwe want to use our theoretical formulae. Actually, we would be measuringthe friction coefficient (permeability).

In one embodiment of the disclosure, porosity of the formation may beestimated. Consider the excitation of the system by stresses

$\left( \frac{\partial u_{z}}{\partial x} \right)$along the surface of the boundary as given by the formula:

$\begin{matrix}{u_{z} = {\frac{\beta_{1}\beta_{2}}{\beta_{1}^{2} - \beta_{2}^{2}}\frac{c_{t}^{2}}{\omega^{2}}{\frac{1 + {{\mathbb{i}}{\overset{\_}{\omega}/\omega_{0}}}}{1 + {{{\mathbb{i}}\left( {1 + ɛ} \right)}{\overset{\_}{\omega}/\omega_{0}}}}\left\lbrack {{\left( {\beta_{2} + \frac{\varphi_{1}}{c_{t}^{2}\beta_{2}}} \right){\mathbb{e}}^{{- \beta_{2}}x}} - {\left( {\beta_{1} + \frac{\varphi_{1}}{c_{t}^{2}\beta_{1}}} \right){\mathbb{e}}^{{- \beta_{1}}x}}} \right\rbrack}{\frac{\partial u_{z}}{\partial x}.}}} & (51)\end{matrix}$On the surface, at resonant frequency, this gives:

$\begin{matrix}{{{u_{z}}_{x = 0} \approx {{- \frac{1}{\beta_{2}}}{\frac{\partial u_{z}}{\partial x}}_{x = 0}}},{{\beta_{2}} \approx {\frac{\omega_{0}}{c_{t}^{2}}{\sqrt{\frac{\rho_{0}}{\rho_{0,s}}}.}}}} & (52)\end{matrix}$Consequently, we get a formula which enables us to determine porosity ifwe have measured the amplitude of the surface oscillations velocity andits given spatial derivative (the tangential force of the surface impactis fixed).

$\begin{matrix}{\frac{1 - d}{{\left( {\frac{\rho_{0,1}^{f}}{\rho_{0,s}^{f}} - 1} \right)d} + 1} \approx {\frac{\omega_{0}^{2}}{c_{t}^{2}}{\frac{{u_{z}}_{x = 0}^{2}}{{\frac{\partial u_{z}}{\partial x}}_{x = -}^{2}}.}}} & (53)\end{matrix}$In an alternate embodiment of the disclosure, the porosity may beobtained by using a nuclear logging device.

To summarize, in one embodiment of the disclosure, a magnetic fieldhaving components along an interface and parallel to the interface isgenerated by an electromagnet. A transducer generates shear vibrationsthat are parallel to the interface in two orthgonal directions. Atransducer measures the velocity of the displacement of the interface intwo orthogonal directions in the interface. The frequency of themagnetic field is varied until the velocity of the displacement in thetwo directions is the same. This defines a resonance frequency. Usingthe measured values of the velocity at the resonance frequency, themagnitude of the magnetic field and the resonance frequency, it ispossible to determine the formation conductivity, the magneticpermeability of the formation and the electrokinetic constant of theformation. Optionally, by producing a stress along the interface andmaking a measurement of the gradient of the velocity along theinterface, it is possible to determine the porosity of the formation.Optionally, the porosity may be determined using a porosity loggingtool. The estimated resistivity properties and/or porosity of the earthformation are used to estimate the reservoir potential of the formationand in development of the reservoir using known methods.

Implicit in the processing of the data is the use of a computer programimplemented on a suitable machine readable medium that enables theprocessor to perform the control and processing. The machine readablemedium may include ROMs, EPROMs, EAROMs, Flash Memories and Opticaldisks. The determined formation permeabilities may be recorded on asuitable medium and used for subsequent processing upon retrieval of theBHA. The determined formation permeabilities may further be telemetereduphole for display and analysis.

1. A method of estimating a resistivity property of an earth formationcomprising: generating a shear wave in an earth formation at aninterface between a fluid and the formation; generating a staticmagnetic field at the interface having a first direction; varying afrequency of a second magnetic field having a second direction differentfrom the first direction until a resonance condition is satisfied in theformation; and using at least one measurement of motion of the earthformation to provide the estimate of the resistivity property.
 2. Themethod of claim 1 wherein generating the shear wave further comprisesgenerating a pair of shear waves having substantially orthogonalpolarization.
 3. The method of claim 1 wherein the first direction issubstantially orthogonal to the interface and the second direction issubstantially parallel to the interface.
 4. The method of claim 1wherein the resonance condition further comprises a substantial equalityof a velocity of motion of the formation in each of two directionsnormal to the interface, the method further comprising measuring thevelocity of motion of the formation in the two directions.
 5. The methodof claim 1 wherein estimating the value of the resistivity propertyfurther comprises estimating a value of at least one of: (i) a formationconductivity, (ii) a permeability of the formation, and (iii) anelecktrokinetic constant of the formation.
 6. The method of claim 5further comprising: (i) generating a stress along the interface, (ii)measuring a gradient of a velocity of motion of the formation along theinterface, and (iii) estimating a porosity of the formation.
 7. Themethod of claim 5 further comprising using a porosity logging tool forestimating a value of porosity of the formation.
 8. The method of claim1 further comprising generating the shear wave and the magnetic fieldusing a logging tool conveyed in a borehole in the earth formation. 9.An apparatus configured to estimate a resistivity property of an earthformation comprising: a transducer configured to generate a shear wavein an earth formation at an interface between a fluid and the formation;a magnet arrangement configured to generate a static magnetic field atthe interface having a first direction and generate a time varyingmagnetic field at the interface in a second direction different from thefirst direction; a processor configured to: (i) vary a frequency of thetime-varying magnetic field until a resonance condition is satisfied inthe formation; and (ii) use at least one measurement of motion of theearth formation to provide the estimate of the resistivity property. 10.The apparatus of claim 9 wherein the first direction is substantiallyorthogonal to the interface and the second direction is substantiallyparallel to the interface.
 11. The apparatus of claim 9 wherein theresonance condition further comprises a substantial equality of avelocity of motion of the formation in each of two directions normal tothe interface, the apparatus further comprising a plurality of geophonesconfigured to the velocity of motion of the formation in the twodirections.
 12. The apparatus of claim 9 wherein the processor isconfigured to estimate the value of the resistivity property by furtherestimating a value of at least one of: (i) a formation conductivity,(ii) permeability of the formation, and (iii) an elecktrokineticconstant of the formation.
 13. The apparatus of claim 12 furthercomprising: an additional transducer configured to generate a stressalong the interface, and wherein the processor is further configured toestimate a porosity of the formation using a measurement of a gradientof a velocity of motion of the formation along the interface.
 14. Theapparatus of claim 12 further comprising a porosity logging configuredto make a measurement indicative of a value of porosity of theformation.
 15. The apparatus of claim 9 further comprising a loggingtool configured to convey the transducer and the magnet.